Multiparticle Interference of Pairwise Distinguishable Photons

One of the central principles of quantum mechanics is that if there are multiple paths that lead to the same event and there is no way to distinguish between them, interference occurs. It is often assumed that distinguishing information in the preparation, evolution, or measurement of a system is sufficient to destroy interference. However, it is still possible for photons in distinguishable, separable states to interfere due to the indistinguishability of paths corresponding to possible exchange processes. Here we experimentally measure an interference signal that depends only on the multiparticle interference of four photons in a four-port interferometer despite pairs of them occupying distinguishable states.

Figure 1

Graphs for multiparticle exchanges and distinguishability. (a) We inject photons with states labeled $a$, $b$, $c$, $d$ into the inputs of a four-port interferometer and count output coincidences. (b) The coincidence probability $P_{1111}$ is determined by the interference of 4! paths associated with the possible exchange processes leading to one photon per output. A subset of four of these 24 paths, each labeled with a Roman numeral, is shown that covers the possible types of exchange. The dependence of these interfering paths on state distinguishability is determined by overlaps that can be described by a graph model. (c) Vertices on a graph represent the states of the photons and directed edges have a weight given by the overlap of states on the connected vertices, here $⟨a|b⟩$. This may be written in modulus-argument form as $r_{ab}\times e^{i{\phi }_{ab}}$ and $r_{ab}=0(1)$ for (in)distinguishable states. (d) This graph has a weight $⟨a|b⟩\times ⟨b|a⟩=r_{ab}^2$ and describes the dependence of $P_{1111}$ on pairwise distinguishability arising from interference of paths related by pairwise exchange. (e) The interference of paths related by threefold exchange depends on another distinguishing parameter: the triad phase, given by the sum of the overlaps’ arguments: ${\phi }_{abc}={\phi }_{ab}+{\phi }_{bc}+{\phi }_{ca}$. We sum the different contributions for each graph. The clockwise and anticlockwise routes, having conjugate phases, will give $\cos {\phi }_{abc}$ [15]. (f) (i) This fourfold exchange has an associated four-particle phase ${\phi }_{abcd}$ that can be decomposed into triad phases if all states overlap: here ${\phi }_{abcd}={\phi }_{abc}+{\phi }_{acd}$. (ii) If pairs of states are distinguishable (a,c and b,d), the internal edges disappear so $r_{ac}=r_{bd}=0$ and three-particle interference vanishes but four-particle interference persists [26].

Figure 2

Preparation of four photons in pairwise distinguishable states. We use polarization and temporal modes to ensure constant pairwise overlap magnitudes while simultaneously eliminating any three-photon overlaps. The four-particle phase is varied by rotating the polarization of $|d⟩$ in the equator of the Bloch sphere.

Figure 3

Experimental setup. Two KDP crystals pumped by pulses at 415 nm each probabilistically generate pairs of orthogonally polarized photons by SPDC. The pump is separated using long-pass filters (LPF), and the photons in each pair are spatially separated using polarizing beam splitters (PBS). The signal and idler spectra are centered at 830 nm, with bandwidths of 2 nm and 12.5 nm, respectively. In order to realize the wave packets in Fig. 2, we apply additional filtering to the idlers using band-pass filters (BPF) for a bandwidth of around 4.2 nm. Motorized delay stages apply fixed temporal delays, and the polarization states are prepared using sets of quarter- and half-wave plates (QWP and HWP). Then the photons are coupled into single-mode fibers to the interferometer. The free-space bulk-optic quitter uses retroreflecting mirrors to double pass spatial modes through a single balanced dielectric beam splitter, allowing a folded design with straightforward path-length matching and good stability (see the Supplemental Material [28]). Single photon avalanche photodiodes detect photons at the four quitter outputs, and coincidences are recorded using a commercial time tagger with a 4 ns coincidence window.

Figure 4

Counts at the quitter outputs. (a) Output singles when sending $|d⟩$ into the first quitter input and blocking all other inputs. (b) Twofold coincidences when sending states $|d⟩$ and $|a⟩$ into the first and fourth quitter inputs. (c) Threefold coincidences when sending states $|d⟩,|c⟩$, and $|a⟩$ into the first, third, and fourth quitter inputs. (d) Total fourfold coincidences when the emissions from both sources are sent to the quitter. The fitted cosine has a visibility of $3.0\pm 0.5\%$, and the numbers of counts per data point are between 6455 and 6012. (e) By closing one of the SPDC sources, we record the output fourfolds arising from double emissions by the other source. These signals have an average of 2160 and 1740 counts per point. (f) Subtracting the backgrounds in (e) from the fourfold counts in (d) yields the $P_{1111}$ signal showing the expected $-\cos {\phi }_{abcd}$ behavior of Eq. (4). The fitted cosine has a visibility of $10.5\pm 2.3\%$ and counts per point of between 2717 and 2216. Error bars from repeated sweeps are omitted if smaller than the data markers. See Supplemental Material [28] for details of data analysis, background subtraction, and simulations.